a. ramšak* j. mravlje t. rejec* r. Žitko j. bonča *
DESCRIPTION
The Kondo effect in multiple quantum dot systems and deformable molecules. http://gawain.elte.hu/mcrtn/. A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *. *. Department of Physics Faculty of Mathematics and Physics University of Ljubljana. Outline. Conductance - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/1.jpg)
![Page 2: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/2.jpg)
A. Ramšak*J. MravljeT. Rejec*R. ŽitkoJ. Bonča*
The Kondo effect in multiple quantum dot
systems and deformable molecules
Department of PhysicsFaculty of Mathematics and Physics
University of Ljubljana
*
![Page 3: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/3.jpg)
OutlineOutline
(1) Conductance (2) Kondo in a single quantum dot(3) Methods(4) Double quantum dots(5) Triple quantum dots(6) Deformable molecules(7) Center-of-mass motion(8) Summary
![Page 4: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/4.jpg)
gV
12
n
~gateV
~sdV IA
( )gatesd
IG G VV
ConductanceConductance
![Page 5: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/5.jpg)
gV
12
n
~gateV
~sdV IA
( )gatesd
IG G VV
ConductanceConductance
![Page 6: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/6.jpg)
Non-interacting systems: U=0
12
n
0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1( )dot
![Page 7: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/7.jpg)
12
n
0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
122( ) [ ]eGh
dot( ) ( )G
Non-interacting systems: U=0
![Page 8: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/8.jpg)
The Anderson model: U > 0
d U
d
![Page 9: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/9.jpg)
0 0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
122( ) [ ]eGh
0U
![Page 10: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/10.jpg)
0 0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
( )
0 0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
0U
U
or
![Page 11: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/11.jpg)
0 0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
122( ) [ ]eGh
0 0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
0U
U
or
KT T
T
![Page 12: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/12.jpg)
d U d
d
d U
( )
0
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 13: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/13.jpg)
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 14: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/14.jpg)
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 15: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/15.jpg)
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 16: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/16.jpg)
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 17: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/17.jpg)
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 18: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/18.jpg)
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 19: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/19.jpg)
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 20: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/20.jpg)
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 21: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/21.jpg)
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 22: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/22.jpg)
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 23: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/23.jpg)
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 24: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/24.jpg)
d U d
d
d U
( )
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 25: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/25.jpg)
d U d
d
d U
( ) ( )1
2
d d UU
KT U e
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 26: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/26.jpg)
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 27: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/27.jpg)
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 28: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/28.jpg)
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
![Page 29: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/29.jpg)
/ 2d U
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
( / 2, )dA U
![Page 30: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/30.jpg)
1
““Ring system” Ring system”
![Page 31: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/31.jpg)
““OpenOpen system” system”
![Page 32: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/32.jpg)
““OpenOpen system” system”
![Page 33: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/33.jpg)
the GS energy of a the GS energy of a largelarge ring ring system is an universal function system is an universal function of fluxof flux
T. Rejec and A. Ramšak, PRB 68, 033306 (2003); 68 035342 (2003)
IF IF openopen system is Fermi liquid system is Fermi liquid
![Page 34: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/34.jpg)
4 1( 3); elN n N even
1N
![Page 35: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/35.jpg)
4 1( 3); elN n N odd
1N
![Page 36: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/36.jpg)
0/g G G
![Page 37: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/37.jpg)
Linear conductance from Linear conductance from the the ground-state energyground-state energy
0T
See also: J. Favand and F. Mila (Phys. J. 1998); O. Sushkov (PRB 2001); R. Molina et al. (PRB 2003)
![Page 38: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/38.jpg)
Linear conductance from Linear conductance from the the ground-state energyground-state energy
0T
![Page 39: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/39.jpg)
Linear conductance from Linear conductance from the the ground-state energyground-state energy
0T
![Page 40: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/40.jpg)
Aharonov – Bohm ringsAharonov – Bohm rings
Broken time-reversal symmetry
T. Rejec and A. Ramšak, PRB 68, 033306 (2003)
![Page 41: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/41.jpg)
The Kondo effect in a quantum dotThe Kondo effect in a quantum dot
U=0
numerical tests…numerical tests…
![Page 42: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/42.jpg)
The Kondo effect in a quantum dot: finite temperatureThe Kondo effect in a quantum dot: finite temperature
T
U=0 high T
low T
![Page 43: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/43.jpg)
The Kondo effect in a quantum dot: finite temperatureThe Kondo effect in a quantum dot: finite temperature
T
high T
low T
![Page 44: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/44.jpg)
The Kondo effect in a quantum dot: finite temperatureThe Kondo effect in a quantum dot: finite temperature
T
high T
low T
![Page 45: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/45.jpg)
loc max/M S S
21 /t t
( / 2, )dA U Fingerprints of Kondo…Fingerprints of Kondo…
/ 2d U
![Page 46: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/46.jpg)
loc max/M S S
21 /t t
( / 2, )dA U Fingerprints of Kondo…Fingerprints of Kondo…
/ 2d U
![Page 47: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/47.jpg)
21 /t t
( / 2, )dA U Fingerprints of Kondo…Fingerprints of Kondo…
/ 2d U
![Page 48: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/48.jpg)
loc max/M S S
21 /t t
( / 2, )dA U Fingerprints of Kondo…Fingerprints of Kondo…
/ 2d U
![Page 49: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/49.jpg)
Chan et al, Nanotechnology 15, 609 (2004)
Vidan et al, Applied Phys. Lett. 85, 3602 (2004)
Electrostatic gates
QD
Elzerman et al, PRB 67, 16308 (2003)
QD
MultipleMultiple quantum dot systems quantum dot systems
![Page 50: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/50.jpg)
Double quantum dotDouble quantum dot
![Page 51: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/51.jpg)
Double quantum dotDouble quantum dot
2d t
2(b)
0t UV
2(a) t U
212
K(c) 4 2tJ TU
2 12t t t''
2RKKY KJ T
![Page 52: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/52.jpg)
Double quantum dotDouble quantum dot
2d t
2(b)
0t UV
2(a) t U
212
K(c) 4 tJ TU
2 12t t t''
2(d) 0 & ~ : (4) Kondot V U SU
![Page 53: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/53.jpg)
2d t
2(b)
0t UV
2(a) t U
212
K(c) 4 tJ TU
2(d) 0 & ~ : (4) Kondot V U SU
![Page 54: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/54.jpg)
J. Mravlje, A. Ramšak, and T. Rejec, Phys. Rev. B 73, 241305(R) (2006)
Also for finite hibridization: [ (4)] ~ [ (2)]K KT SU U T SU
half filling: 2n
![Page 55: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/55.jpg)
hibridization
![Page 56: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/56.jpg)
Double quantum dotDouble quantum dot
2 Kondo2 Kondo
AFMAFM
1 Kondo1 Kondo
t t
12/ 3U t
12/ 50U t
12/ 60U t
1 2S S
![Page 57: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/57.jpg)
Double quantum dotDouble quantum dot
2 Kondo2 Kondo
AFMAFM
1 Kondo1 Kondo
t t
12/ 3U t
12/ 50U t
12/ 60U t
1 2S S
![Page 58: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/58.jpg)
Double quantum dotDouble quantum dot
2 Kondo2 Kondo
AFMAFM
1 Kondo1 Kondo
t t
12/ 3U t
12/ 50U t
12/ 60U t
1 2S S
![Page 59: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/59.jpg)
Other topologies: local singlet vs the Kondo effect
A. Ramšak, J. Mravlje, R. Žitko, and J. Bonča, quant-ph/0608065.
![Page 60: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/60.jpg)
Thermal equilibrium: A-B spin corelations
/B J/T J
2
4tJU
A BS S
![Page 61: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/61.jpg)
2
4tJU
Zero magnetic field, thermal equilibrium
A
B
0, .B T const
![Page 62: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/62.jpg)
2
4tJU
/J U
Zero magnetic field, temperature
A
B
![Page 63: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/63.jpg)
2
4tJU
/J U
A
B
Zero magnetic field, temperature
![Page 64: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/64.jpg)
/J U2 cJ
A
B
Zero magnetic field, temperature
![Page 65: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/65.jpg)
/J U
3 ~cJ
A
B
Zero magnetic field, temperature
![Page 66: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/66.jpg)
Triple quantum dotTriple quantum dot
d
22d t
22d t
![Page 67: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/67.jpg)
d
22d t
22d t
d U
Triple quantum dotTriple quantum dot
![Page 68: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/68.jpg)
d
Triple quantum dotTriple quantum dot
![Page 69: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/69.jpg)
d
Triple quantum dotTriple quantum dot
![Page 70: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/70.jpg)
Triple quantum dotTriple quantum dot
d
![Page 71: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/71.jpg)
2/ 1U t
10
20
Triple quantum dotTriple quantum dot
2
5
![Page 72: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/72.jpg)
Triple quantum dotTriple quantum dot
1~ Kt T
![Page 73: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/73.jpg)
Triple quantum dotTriple quantum dot
1~ Kt T
![Page 74: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/74.jpg)
Triple quantum dotTriple quantum dot
1~ Kt T
~t J
![Page 75: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/75.jpg)
Triple quantum dotTriple quantum dot
1~ Kt T
~t J
1~ 2 KJ T
![Page 76: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/76.jpg)
Triple quantum dotTriple quantum dot
![Page 77: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/77.jpg)
Deformable molecules Deformable molecules
e
![Page 78: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/78.jpg)
Deformable molecules Deformable molecules
e
![Page 79: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/79.jpg)
Deformable molecules Deformable molecules
H. Park et al. Nature 407 (2000)
e
Change in:• local energy• hopping matrix elements
![Page 80: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/80.jpg)
…
![Page 81: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/81.jpg)
Modeling Modeling
![Page 82: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/82.jpg)
))(1( aanMaanUnnH d
2eff (2) 2 (1) 2U E E U M
Isolated molecule:
d
d
molecule attached to the leads: Lang & Firsov transformation:
)1)((1 ,~
naaMeUHUUH
the result:
,M
,M
Reduction of U and narrowing of the level-widthA.C. Hewson and D.M. News J.Phys C 13 (1980)K. Schönhammer and O. Gunnarsson PRB 30 (1984)
Old knowledge …Old knowledge …
![Page 83: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/83.jpg)
Udecrease
negative U:A. Taraphder and P. Coleman,PRL 66, 2814 (1991).
![Page 84: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/84.jpg)
M
J. Mravlje, A. Ramšak, and T. Rejec, PRB 72, 121403(R) (2005);See also: P.S. Cornaglia, D.R. Grempel, and H. Ness, Phys. Rev. B 71, 075320 (2005), A. Mitra, I. Aleiner, and A.J. Millis, Phys. Rev. B 79, 245302 (2004).
![Page 85: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/85.jpg)
Molecules with a center of mass motionMolecules with a center of mass motion
J. Mravlje, A. Ramšak, and T. Rejec, submitted to PRB
![Page 86: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/86.jpg)
Molecules with a center of mass motionMolecules with a center of mass motion
![Page 87: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/87.jpg)
Molecules with a center of mass motionMolecules with a center of mass motion
![Page 88: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/88.jpg)
Molecules with a center of mass motionMolecules with a center of mass motion
![Page 89: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/89.jpg)
Molecules with a center of mass motionMolecules with a center of mass motion
Friedel sum rule:
![Page 90: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/90.jpg)
Molecules with a center of mass motionMolecules with a center of mass motion
Friedel sum rule:
![Page 91: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/91.jpg)
Molecules with a center of mass motionMolecules with a center of mass motion
A
B
![Page 92: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/92.jpg)
Molecules with a center of mass motionMolecules with a center of mass motion
![Page 93: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/93.jpg)
Molecules with a center of mass motionMolecules with a center of mass motion
![Page 94: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/94.jpg)
Summary Summary
• Linear conductance at T=0 can then be extracted from the GS energy:
• The Kondo effect: Low temperature destiny of quantum dots
![Page 95: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/95.jpg)
U t
d
U
d
t0/g G G
ad summary…
![Page 96: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/96.jpg)
Formulae are exact IF the system is Fermi liquid
note:
• linear conductance• zero temperature• non-interacting single-channel leads
![Page 97: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/97.jpg)
Fisher – Lee relation …
Conductance formalismsConductance formalisms
non-equilibrium transport: T ≠ 0, V ≠ 0
U = 0
Landauer – Büttiker formula
linear response regime: T ≠ 0, V ~ 0
zero-temperature linear response: T = 0, V ~ 0
U ≠ 0
Meir – Wingreen formula
In Fermi liquid systems
Kubo formula
![Page 98: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/98.jpg)
Proof of the method
Step 1. Conductance of a Fermi liquid system at T=0
Kubo
T=0
define (n.i.: Fisher-Lee)
‘Landauer’
![Page 99: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/99.jpg)
Step 2. Quasiparticle hamiltonian (Landau Fermi liquid)
![Page 100: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/100.jpg)
Step 3. Quasiparticles in a finite system
N
![Page 101: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/101.jpg)
Step 4. Validity of the conductance formulas
![Page 102: A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča *](https://reader036.vdocuments.mx/reader036/viewer/2022062501/56816057550346895dcf80c3/html5/thumbnails/102.jpg)